# Semi-Empirical Model of Remote-Sensing Reflectance for Chosen Areas of the Southern Baltic

^{1}

^{2}

^{*}

## Abstract

**:**

_{rs}(λ

_{i})) for four wavelengths (λ

_{i}= 420 nm, 488 nm, 555 nm, and 620 nm) based on multiparameter algorithms of absorption (a(λ

_{i})) and backscattering (b

_{b}(λ

_{i})) coefficients. The bio-optical properties of water were determined based on empirical data gathered from aboard the r/v Oceania from April 2007 to March 2010 in chosen areas of the southern Baltic (Polish coast). The analyses reveal that R

_{rs}(λ

_{i}) in the studied area can be described with satisfactory accuracy using a five-parameter model. Positive results with a statistical error magnitude of R

_{rs}(λ

_{i}) of less than 50% were achieved for all four applied wavelengths. Bio-optical algorithms proposed by the authors enable evaluating biogeochemical characteristics of coastal areas in a broader context of ecosystem assessment and contribute significantly to the development of Earth and environmental sciences.

## 1. Introduction

#### 1.1. Context of the Study

_{rs}) is a crucial parameter in optical oceanography and is often used for the development of algorithms to estimate bio-optical components of seawater. Diffusion of light upon interaction with matter depends on the types and concentrations of the water components. Suspended and dissolved matter changes the optical properties of water, particularly in bays and coastal areas. The coastal waters of the Gulf of Gdańsk (the southern part of the Baltic Sea, Poland) are areas of high diversity and dynamic changes in physical and chemical properties of water, making this area an ideal location to study optical properties of water. As a typical coastal environment, these waters are characterised by specific physical conditions such as limited light penetration and high rates of transport and sedimentation of organic suspended particulate matter (SPM

_{org}) and inorganic suspended particulate matter (SPM

_{inorg}). There are many algorithms for estimating surface concentrations of suspended particulate matter (SPM) based on remote-sensing measurements [1,2]. However, most of them have been developed for the so-called “Case 1 waters” [3], i.e., areas where phytoplankton is the main factor responsible for variations in optical properties of the water (mostly open oceans) [4]. On the other hand, coastal waters and bays known as Case 2 waters [3] are influenced not just by phytoplankton but also by other substances that vary independently of phytoplankton SPM

_{inorg}and coloured dissolved organic matter (CDOM) [5,6,7]. Therefore, to apply remote sensing in such areas, their variability must be understood. The extraordinarily high optical diversity of Case 2 waters makes interpretation of optical signal from these waters rather difficult and more complex in terms of composition and optical properties than in Case 1 waters. The problem described also applies to the coastal waters of the Gulf of Gdańsk, where the correlation between optical properties and concentrations of optically active components is strictly local.

#### 1.2. State of Knowledge

_{rs}for 555 nm/440 nm for dispersed oil detection using any optical parameters [15]. The impact of absorption coefficient (a(λ

_{i})) and backscattering coefficient b

_{b}(λ

_{i}) on R

_{rs}(λ

_{i}) is highly variable, thus the interpretation of reflectance spectra requires a simultaneous multi-parameter analysis of light propagation in seawater [16,17].

_{i}) and b

_{b}(λ

_{i})) and apparent optical properties (AOPs) (e.g., R

_{rs}(λ

_{i})) and also statistical relationships between the IOPs and the biogeochemical components of water. Such algorithms are typically used to describe optically complex Case 2 waters [18,19,20,21,22,23,24,25]. However, because they show a very high sensitivity to changes in the composition and concentration of various water components, their range of applicability is limited and they can only be used locally.

#### 1.3. Research Objectives

_{rs}(λ

_{i}), which is easily measurable and forms a basis for remote-sensing methods, is (under certain conditions of external lighting such as optical state of the atmosphere, cloud cover, position of the sun in the sky, etc.) closely related to a(λ

_{i}) and b

_{b}(λ

_{i}). Hence, the knowledge of all these interdependencies, especially the determination of their mathematical quantitative descriptions, is extremely important in the development of remote-sensing methods of controlling the condition and functioning of marine ecosystems (e.g., via satellites).

- determination of a mathematical description of the relationship between the selected optical properties (absorption coefficient of phytoplankton (a
_{ph}(λ_{i})), absorption coefficient by non-algal particles (a_{d}(λ_{i})), the coloured dissolved organic matter absorption coefficient (a_{CDOM}(λ_{i})), backscattering coefficient of particles (b_{bp}(λ_{i}))) and the concentrations and physicochemical properties of natural water components (Chl a, SPM, surface concentrations of absorption coefficients of coloured dissolved organic matter for wavelength 400 nm (a_{CDOM}(400)), sum of surface concentrations of accessory pigments (∑C), SPM_{inorg}) in selected areas of the southern Baltic; - development of a semi-empirical model of R
_{rs}(λ_{i}) enabling the determination of R_{rs}(λ_{i}) spectra in the visible light range based on the known spectra of a(λ_{i}) and b_{b}(λ_{i}) in coastal waters of the southern Baltic or based on knowledge of the concentration of admixture components.

_{b}and particle size distribution in Case 2 waters.

## 2. Materials and Methods

#### 2.1. The Conception of the Five-Parameter Model of R_{rs}

_{rs}consisting of three sections: A—input data, B—model formula and C—calculations. Before starting work on the model, we checked the correlations between IOPs (a

_{ph}, a

_{d}, a

_{CDOM}and b

_{bp}) and the biogeochemical constituents (including organic fraction of SPM

_{org}). We chose the best correlations between biogeochemical components and optical properties for our work. This paper presents the strongest correlations between the aforementioned parameters.

_{i}= 420 nm, 488 nm, 555 nm and 620 nm). First, regression methods of non-linear functions of one variable were used to designate 1-parameter statistical algorithms of b

_{bp}: b

_{bp}(λ

_{i}) = f(SPM). This allowed us to determine values of this optical parameter based on the predetermined surface concentrations of SPM. Similar one-parameter statistical analyses were conducted for other measured optical properties: a

_{ph}, a

_{d}, a

_{CDOM}(a

_{ph}(λ

_{i}) = f(Chl a), a

_{d}(λ

_{i}) = f(SPM), a

_{CDOM}: a

_{CDOM}(λ

_{i}) = f(a

_{CDOM}(400))) and concentrations of biogeochemical (and optical) constituents of waters-Chl a, SPM, a

_{CDOM}(400).

_{CDOM}is complex due to the variety of chemical structures of substances dissolved in natural waters and differentiated interaction of their molecules with electromagnetic radiation. They include both saturated substances (which practically do not absorb light in the uv visible spectrum) and strongly absorbant unsaturated substances. It would be pointless to search for any dependency between a

_{CDOM}and SPM. Therefore, the “optical” indicator of the concentrations of CDOM substances we adopted is a

_{CDOM}for the selected reference wavelength a

_{CDOM}(λ

_{ref}), which has been used commonly for many years now [4,31,37,38,39]. We assumed that λ

_{ref}= 400 nm, because λ

_{ref}is mostly located in the violet-blue region of the spectrum.

_{bp}: b

_{bp}= f(SPM, SPM

_{inorg}), that allowed us to calculate the value of b

_{bp}, based on e.g., SPM and SPM

_{inorg}. Similar multi-parameter algorithms were employed for other optical properties and water constituents. We developed a statistical two-parameter model of a

_{ph}: a

_{ph}(λ

_{i}) = f(Chl a, ∑C) and a statistical two-parameter model of a

_{d}: a

_{d}(λ

_{i}) = f(SPM, SPM

_{inorg}). a(λ

_{i}) was calculated based on the equation [40]:

_{i}) = a

_{ph}(λ

_{i}) + a

_{d}(λ

_{i}) + a

_{CDOM}(λ

_{i}) + a

_{w}(λ

_{i}),

_{w}(λ

_{i}) is the absorption coefficient of seawater molecules [41]. b

_{b}(λ

_{i}) was estimated as [40]:

_{b}(λ

_{i}) = b

_{bp}(λ

_{i}) + b

_{bw}(λ

_{i}),

_{bw}(λ) is the backscattering of seawater molecules according to Morel [42].

_{i}values for the four tested wavelengths, λ

_{i}= 420, 488, 555 and 620 nm.

_{rs}: R

_{rs}(λ

_{i}) = f(Chl a, ∑C, SPM, SPM

_{inorg}, a

_{CDOM}(400)) for the selected areas of the southern Baltic, based on multi-parameter algorithms for a(λ

_{i}) and b

_{b}(λ

_{i}).

#### 2.2. The Study Area

#### 2.3. Data Acquisition and Processing

_{bp}(λ

_{i}) were estimated based on in situ measurements in the near-surface layer (1 m depth) using a spectral backscattering meter Hydroscat-4 (HOBI Labs, Bellevue, Washington, WA, USA) at four wavelengths (λ

_{i}= 420, 488, 550 and 620 nm). Furthermore we used the radiometer Ocean Colour Profiler OCP–100 (Satlantic Inc., Victoria, BC, Canada) to estimate R

_{rs}(λ

_{i}) from the in situ measurements. Next, the values of R

_{rs}(λ

_{i}) calculated using only the OCP-100 were used to calculate the f/Q ratio.

_{inorg}, and ∑C were obtained, and the following optical parameters of the water were measured: a

_{ph}(λ

_{i}), a

_{d}(λ

_{i}), a

_{CDOM}(λ

_{i}). During vessel cruises filtration of water samples was conducted right after collection. In the research trips to the pier in Sopot, filtration was performed a few hours after the sampling.

_{org}and SPM

_{inorg}was calculated using the standard gravimetric technique [25,50,51]. The concentration of SPM was measured gravimetrically after filtration of the same amount of water through pre-weighed and pre-combusted filters; next, the inorganic fraction was weighed after combustion. The organic fraction was determined by subtracting SPM

_{inorg}from SPM.

_{CDOM}(λ

_{i}) and a

_{CDOM}(400) for spectrophotometric analysis used two-step filtration and appropriate storage of samples awaiting laboratory analysis (at 4 °C, for no longer than 3 weeks) [57,58]. The first step of filtration eliminates large particles of suspended matter. To this purpose, a GF/F filter (by Whatman, with a pore size of 0.7 μm) was applied. The second stage was the removal of the smallest suspended particles using cellulose membrane filters (Sartorius) (pore size 0.2 μm). The spectra of a

_{CDOM}(λ

_{i}) were measured with a spectrophotometer in a 10 cm cuvette relative to Milli-Q water using samples filtered through a pre-rinsed 0.2 μm filter [59]. The total particulate absorption −a

_{p}(λ

_{i}), was measured with a spectrometer employing the Whatman GF/F filter pad technique [60] followed by depigmentation with sodium hypochloride, which separates a

_{ph}(λ

_{i}) and a

_{d}(λ

_{i}) components. a

_{ph}(λ

_{i}) was calculated as the difference between a

_{p}(λ

_{i}) and a

_{d}(λ

_{i}):

_{ph}(λ

_{i}) = a

_{p}(λ

_{i}) − a

_{d}(λ

_{i}).

_{b}(λ

_{i}) was measured in situ using backscattering meter Hydroscat-4 (HOBI Labs) at four different wavelengths: 420 nm, 488 nm, 555 nm, 620 nm. The methodology of b

_{b}(λ

_{i}) measurements, calibration procedures, and subsequent determination of b

_{b}(λ

_{i}) have been described by Maffione and Dana [61]. To estimate b

_{b}(λ

_{i}) [61,62] we used values of volume scattering function $\left(\beta \left(\Psi \right)\right)$ at an angle of 140

^{o}. We took data from absorption and attenuation meter ac-9 (WET Labs, Philomath, OR, USA) to a procedure called sigma correction (a correction for an incomplete recovery of backscattered light in highly attenuating waters, according to the User’s Manual [63]).

_{rs}(λ

_{i}) was also calculated using data obtained from radiometer OCP-100 (just below the water). The R

_{rs}(λ

_{i}) values estimated by OCP-100 were used to calculate the f/Q ratio. OCP-100 measured upwelling radiance (L

_{u}(λ

_{i})) and downwelling irradiance (E

_{d}(λ

_{i})) on seven channels: λ

_{i}= 412, 443, 490, 510, 555, 670 and 683 nm. We used nearest-neighbour interpolation to estimate wavelengths L

_{u}(420), E

_{d}(420), L

_{u}(488), E

_{d}(488), L

_{u}(620) and E

_{d}(620), between the measured data wavelengths. The L

_{u}(λ

_{i}) values obtained from the OCP-100 m were corrected for self-shading effects [64,65]. The values of R

_{rs}(λ

_{i}) were calculated using the following equation:

^{−}) means just below the water.

_{r}(λ

_{i})

_{s}is also proportional to the ratio of b

_{b}and a based on Morel and Gentili [66]. In our study we used the following equation [66]:

_{u}(0

^{–})) is upwelling irradiance.

_{rs}values has been the subject of many theoretical analyses [68]. In 2002, Morel et al. [69] established that for Case 2 water, the f/Q ratio ranges from 0.07 to 0.18 sr

^{−1}, and it depends on the concentration of Chl a and the illumination of the area (e.g., solar zenith angle, waves, etc.). These theoretical analyses were supported by empirical research conducted by Voss and Morel in 2005 [70]. However, for Case 2 waters, determining the f/Q ratio is not easy and requires a strictly local approach. In 2003, D’Sa and Miller [35] determined that at the mouth of the Mississippi River in the Gulf of Mexico, this ratio ranges from 0.09 to 0.12 sr

^{−1}. We attempted to determine the empirical f/Q values for the analysed areas and established that the f/Q ratio for the southern Baltic water ranges from 0.07 to 0.13 sr

^{−1}and depends on the wavelength (it increases with the wavelength). For this purpose, we used measurements of L

_{u}(λ

_{i}), E

_{d}(λ

_{i}), b

_{b}(λ

_{i}) and a(λ

_{i}) for four wavelengths of light (λ

_{i}= 420, 488, 555 and 620 nm). Next, using Equations (4) and (5), we calculated f/Q values for the four tested wavelengths.

## 3. Results

#### 3.1. Analysis of the Impact of Biogeochemical Components on the Optical Properties of the Southern Baltic Coastal Waters

_{ph}(λ

_{i}), a

_{d}(λ

_{i}), a

_{CDOM}(λ

_{i}) and b

_{bp}(λ

_{i})) and concentration of a single parameter (Chl a, SPM and a

_{CDOM}(400)) were statistically approximated for all four wavelengths. Figure 3 shows the relationships between the empirically determined a

_{ph}(λ

_{i}), a

_{d}(λ

_{i}), a

_{CDOM}(λ

_{i}) (Figure 3a–c) and b

_{bp}(λ

_{i}) (Figure 3d) for a wavelength 488 nm and concentration of biogeochemical components of water (Chl a, SPM and a

_{CDOM}).

_{ph}(488), a

_{d}(488), a

_{CDOM}(488), and b

_{bp}(488) and the concentration of biogeochemical components. In Figure 3a, we see that correlation between a

_{ph}(488) and Chl a is better than the relationships: a

_{d}(488)-SPM, a

_{CDOM}(488)-a

_{CDOM}(400), and b

_{bp}(488)-SPM. This relationship is also the most linear in contrast to the correlation between a

_{CDOM}(488) and a

_{CDOM}(400), which was the nature of absorption by CDOM is particularly complex due to the variety of chemical structures of dissolved substances in natural waters and the differentiation of the interaction of their molecules with electromagnetic radiation. They include both saturated substances, which practically do not absorb light in the visible range, and unsaturated substances, which strongly absorb light. Therefore, in the a

_{CDOM}analyses, we adopted a

_{CDOM}(400) commonly used for many years also by other authors [4,37,71], namely the a

_{CDOM}for the selected reference wavelength, a

_{CDOM}(λ

_{ref}). We assumed that λ

_{ref}= 400 nm.

_{d}(λ

_{i}), a

_{ph}(λ

_{i}), and a

_{CDOM}(λ

_{i}) in the chosen areas of the southern Baltic. We can see that a

_{CDOM}has the greatest contribution in the total absorption at a wavelength of 420 nm and its average percentage is 68%, while the average shares of other absorption coefficients are a

_{ph}-20% and a

_{d}-12%.

_{ph}(λ

_{i}) and Chl a (Equations (7)–(10)) and also between a

_{d}(λ

_{i}) and SPM (Equations (11)–(14)), or between b

_{bp}(λ

_{i}) and SPM (Equations (15)–(18)) are well approximated by power functions such as:

_{ph}(420)

_{cal}= 0.056(Chl a)

^{0.827},

_{ph}(488)

_{cal}= 0.037(Chl a)

^{0.820},

_{ph}(555)

_{cal}= 0.013(Chl a)

^{0.815},

_{ph}(620)

_{cal}= 0.008(Chl a)

^{0.926},

_{d}(420)

_{cal}= 0.071(SPM)

^{0.809},

_{d}(488)

_{cal}= 0.045(SPM)

^{0.762},

_{d}(555)

_{cal}= 0.031(SPM)

^{0.646},

_{d}(620)

_{cal}= 0.002(SPM)

^{0.592},

_{bp}(420)

_{cal}= 0.011(SPM)

^{0.911},

_{bp}(488)

_{cal}= 0.008(SPM)

^{0.891},

_{bp}(555)

_{cal}= 0.007(SPM)

^{0.935},

_{bp}(620)

_{ca}

_{l}= 0.005(SPM)

^{0.881}.

_{CDOM}(λ

_{i}) and a

_{CDOM}(400) (Equations (19)–(22)) are well approximated by second order non-linear exponential functions:

_{d}(λ

_{i}), a

_{ph}(λ

_{i}), and a

_{CDOM}(λ

_{i}) and b

_{bp}(λ

_{i}) coefficients with the values of the coefficients calculated based on Equations (7)–(22) (Figure 5). In addition, statistical errors were determined (Table 1).

_{i,m}—measured values; X

_{i,cal}—estimated values [the subscript m stands for “measured”; cal—stands for “calculated”]):

- relative mean error (systematic error):

- 2.
- standard deviation (statistical error) of ε (RMSE-root mean square error):

- 3.
- mean logarithmic error:

- 4.
- standard error factor:

- 5.
- statistical logarithmic errors:

- 6.
- ${\sigma}_{log}-standarddeviationofthesetlog\left(\frac{{X}_{i,cal}}{{X}_{i,m}}\right);$

- 7.
- $\langle log\left(\frac{{X}_{i,cal}}{{X}_{i,m}}\right)\rangle -meanoflog\left(\frac{{X}_{i,cal}}{{X}_{i,m}}\right).$

_{p}

_{h}quoted in Table 1 indicate fairly accurate selection of the approximating functions. Similar functions approximating the relationship a

_{ph}(λ

_{i}) = f(Chl a) were used for Case 1 waters by Bricaud et al. [5], and for the waters of the Baltic Sea by Woźniak, Meler et al. [2,31,32,72].

_{d}(λ

_{i}), the error values amount to several dozen or even exceed one hundred per cent. This can be explained partly by methodological reasons. Measurements of a

_{d}(λ

_{i}) and SPM are very complex and subject to significant errors. There is a large variety of SPM in Case 2 waters, which translates into different density and absorption capacities. This is undoubtedly a significant cause of high errors in estimating these coefficients based solely on the total weight of the suspended matter. The non-algal particles consist of organic detritus and mineral particles. Therefore, further analyses took into account the types of suspended particles, i.e., their organic and inorganic fractions.

_{CDOM}(λ

_{i}), the approximation errors are relatively low and amount to a few per cent for the coefficients relating to the wavelength of 420 nm, i.e., the closest reference wave (Table 1). Moving further from this wavelength, the errors of the estimated a

_{CDOM}(λ

_{i}) increase, reaching several per cent for λ

_{i}= 488 nm, about 30% for λ

_{i}= 555 nm, and nearly 40% for λ

_{i}= 620 nm. The errors in the one-parameter estimation for a

_{CDOM}(λ

_{i}) presented above indicate that its accuracy is satisfactory. Therefore, we found that the determined model descriptions of a

_{CDOM}(λ

_{i}) (Equations (19)–(22)) can be used successfully in the model of the reflectance coefficient for selected areas of the southern Baltic.

_{bp}(λ

_{i}) determination based on the SPM obtained using Equations (15)–(18) (Table 1) is definitely better than the accuracy of determining a

_{d}(λ

_{i}) based on the analogous dependence of these coefficients on the SPM. In the case of absorption, these errors are usually twice as great as in the case of scattering. Given that in the natural environment the values of b

_{bp}(λ

_{i}) as well as a

_{ph}(λ

_{i}) vary over several orders of magnitude, error values mostly in the range from about 30% to about 50% for b

_{b}(λ

_{i}of the suspended particles are acceptable. However, we have attempted to improve the accuracy of the estimated b

_{bp}(λ

_{i}) and a

_{ph}(λ

_{i}) values by including additional biogeochemical parameters: SPM

_{inorg}for b

_{bp}(λ

_{i}) and ∑C for a

_{ph}(λ

_{i}) (as in the case of a

_{d}(λ

_{i})).

_{ph}-∑C, and in the case of a

_{d}and b

_{bp}-SPM

_{inorg}(Figure 6).

_{ph}(λ

_{i}), a

_{d}(λ

_{i}) and b

_{bp}(λ

_{i}) values determined by Equations (8), (12) and (16), to the measured values of a

_{ph}(λ

_{i}), a

_{d}(λ

_{i}) and b

_{bp}(λ

_{i}) was compared with the ratio of of Chl a and ∑C (for a

_{ph}(λ

_{i})) and the ratio of SPM and SPM

_{inorg}(for a

_{d}(λ

_{i}) and b

_{bp}(λ

_{i})) (Figure 6, Equations (27)–(29)).

_{ph}(488)

_{m}, a

_{d}(488)

_{m}and b

_{bp}(488)

_{m}with their respective values estimated using the two-parameter model of a

_{ph}(488)

_{cal}, a

_{d}(488)

_{cal}and b

_{bp}(488)

_{cal}and ratio distribution histograms (a

_{ph}(488)

_{cal}/a

_{ph}(488)

_{m}) is presented in Figure 7. Statistical errors (according to linear and logarithmic statistics) of these estimates were also determined (Table 2).

_{ph}(λ

_{i}) on ∑C and Chl a are more accurate than the dependence of a

_{ph}(λ

_{i}) on the concentration of only one parameter (see Table 1, Equations (31)–(34)). The differences between the errors determined for both obtained relationships are in the range from 0.2% to approximately 6.5%. They are respectively equal for the successive wavelengths: (1) for arithmetic statistics −2.1%, 3.8%, 1.6% and 0.2% and (2) for logarithmic statistics −2.5%, 6.4%, 1.1% and 0.6%. Thus, the model described by Equations (31)–(34), taking into account two parameters (∑C and Chl a), is more suitable for the determination of a

_{ph}(λ

_{i}).

_{d}(Table 2, Equations (35)–(38)), systematic errors are relatively small compared to statistical errors. The standard error factor determined for the model of a

_{d}(λ

_{i}) is characterised by a similar tendency as the model error factor a

_{ph}(λ

_{i}) (Table 1, Equations (31)–(34)), i.e., its value increases with increasing wavelength. For λ

_{i}= 420 nm, it is x = 1.91, while for λ

_{i}= 620 nm, it is higher by almost a half and amounts to x = 2.67. As in the case of a

_{ph}(λ

_{i}), we observe an improvement in the a

_{d}(λ

_{i}) estimations after introducing an additional parameter. Thus, also in this model, the dependency of a

_{d}(λ

_{i}) on the SPM and SPM

_{inorg}is better than the dependence of these coefficients on the concentration of only one parameter, SPM.

_{bp}(λ

_{i}) statistical error values (estimated based on SPM and SPM

_{inorg}—Equations (39)–(42)) ranges from 30% to 50% (Table 2) and is slightly smaller than the error magnitudes of b

_{bp}(λ

_{i}) estimated based on SPM—Equations (15)–(18) (Table 1). The differences between these error magnitudes range from about 0% to 9% depending on the wavelength. Additionally, they are also different in the case of arithmetic and logarithmic statistics. In the arithmetic statistics they are close to 0% (for the wavelengths 420, 488, and 555 nm) or approximately 2% (for the light with a wavelength of 620 nm), while logarithmic statistics they are approximately 0.7% for 420 nm, 4.3% for 488 nm, 6.2% for 555 nm, and 8.9% for 620 nm. This means that the variation in the chemical nature of the suspended particles affects their optical scattering capacity only slightly, just as it happened in the case of the a

_{d}albeit to a lesser extent.

#### 3.2. The Five-Parameter Semi-Empirical Model of R_{rs}(λ_{i}) of the Southern Baltic Coastal Waters

_{rs}(λ

_{i}) in selected sea areas of the southern Baltic, which allows us to calculate R

_{rs}(λ

_{i}) for four wavelengths in the visible light range. For this purpose, we used Equation (5) and we based on the known spectra of a(λ

_{i}) and b

_{b}(λ

_{i}), or known concentrations of biogeochemical constituents occurring in these waters (Equations (43)–(46), Table 3).

#### 3.3. Assessment of Estimation Errors of the Five-Parameter R_{rs}(λ_{i}) Model

_{rs}(λ

_{i})

_{cal}/R

_{rs}(λ

_{i})

_{m}is shown in Figure 8. Moreover, the figure presents a graphical comparison of the empirical values of R

_{rs}(λ

_{i})

_{m}with values calculated using the R

_{rs}(λ

_{i})

_{cal}model.

_{rs}model, the dispersion of points is significant (especially for the wavelength of 620 nm) and in all graphs, it takes a very similar form. The greater scatter of points for the 620 nm wavelength can be explained by the fact that this region of the spectrum is especially influenced by the processes of light absorption by SPM (both organic and non-organic particles). The water areas in which the research was conducted are centres very rich in both dissolved and suspended matter.

_{rs}(λ

_{i}) values calculated in the five-parameter model do not exceed 50% for all wavelengths (Table 4). The result is satisfactory since the coastal zone of the southern Baltic is the area characterized by high water dynamics and values of physicochemical parameters can be high, especially in the areas of river mouths. Thus, in such conditions it is extremely difficult to find bio-optical algorithms that would enable accurate estimation of optical quantities based on e.g., the concentration of various types of optically active dopant.

## 4. Discussion

#### 4.1. Reference to the Main Research Objectives

_{ph}(λ

_{i}), a

_{d}(λ

_{i}), a

_{CDOM}(λ

_{i}), b

_{bp}(λ

_{i})) and AOPs (R

_{rs}(λ

_{i})) and the concentrations of biochemical components of water (Chl a, ∑C, SPM, SPM

_{inorg}, a

_{CDOM}(400)) in selected Case 2 waters.

_{bp}(λ

_{i}) and in vitro (a

_{ph}(λ

_{i}), a

_{d}(λ

_{i}), a

_{CDOM}(λ

_{i}), a

_{CDOM}(400), Chl a, ∑C, SPM, SPM

_{inorg}). It was also necessary to carry out a theoretical analysis of a(λ

_{i}) and b

_{b}(λ

_{i}) and to establish the relationship between these optical properties and biogeochemical parameters of seawater components.

#### 4.2. Summary of the Main Findings of the Article

_{b}(λ

_{i}) by suspended matter and the concentrations of these components in the water were established. Moreover, model descriptions of a(λ

_{i}) and b

_{b}(λ

_{i}) for selected waters of the southern Baltic were developed, allowing us to calculate a(λ

_{i}) and b

_{b}(λ

_{i}) for the four selected wavelengths, based on the known concentrations of these components in the waters. Finally, as a result of the analyses conducted, a set of mathematical equations for calculating R

_{rs}(λ

_{i}) spectra in the visible light range (for wavelengths λ

_{i}= 420, 488, 555, and 620 nm) was developed. The formulas are based on the known spectra of a(λ

_{i}) and b

_{b}(λ

_{i}) in the coastal waters of the southern Baltic or based on the knowledge of the concentration of admixture components present there.

_{rs}(λ

_{i}) model from a nearshore location in the Baltic Sea. Their algorithms were based on empirical data of IOPs. They used the Hydrolight model to calculate R

_{rs}.

#### 4.3. Limitations of Our Research

#### 4.4. Recommendations for Future Research

## 5. Conclusions

_{ph}(λ

_{i}) on the concentrations of Chl a, it was found that a

_{ph}(λ

_{i}) increases with a rise in the values of Chl a concentrations. However, these values are not directly proportional to each other, as the increase in a

_{ph}(λ

_{i}) is lower than the increase in Chl a concentrations. These regularities are described well by the power function (Equations (7)–(10)).

_{ph}(λ

_{i}) in chosen areas of the southern Baltic using model descriptions of these coefficients as a function of two independent variables (see Equations (31)–(34)), i.e., Chl a and ∑C, is better than estimates of a

_{ph}(λ

_{i}) as a function of Chl a. The calculated differences between the estimation errors for both of these model descriptions are approximately 2.5%, ranging from about 1% to about 5%, and they depend on the wavelength (see Table 1 and Table 2).

_{d}(λ

_{i}) values on the total concentrations of SPM in seawater indicated that these coefficients rise with an increase in suspended matter concentration, although the rise is not linear. The increase in a

_{d}(λ

_{i}) is less intense than the increase in the total suspended matter concentration. These regularities are described by hyperbolic functions (see Equations (11)–(14)). The statistical analyses performed showed that the accuracy of the estimation of a

_{d}(λ

_{i}) increases when they are described as functions of two independent variables (i.e., SPM and SPM

_{inorg}of the suspended matter fraction). These tendencies are described well by hyperbolic functions (see Equations (35)–(38)). The calculated differences between the estimation errors for both of these model descriptions ranging from about small negative values (i.e., when the accuracy of the estimation based on a function with one independent variable is better than the accuracy of using the dependence on two variables) to about 12.4% and they also depend on the wavelength (see Table 1 and Table 2).

_{CDOM}(λ

_{i}) in seawater rise with an increase in the value of the optical index of their concentrations, a

_{CDOM}(400). They can be described with satisfactory accuracy as a hyperbolic function of a single variable (see Equations (19)–(22)). In the case of a

_{CDOM}, the approximation errors are relatively low for short waves (a few percent for λ = 420 nm) and they increase with the wavelength approaching 40% for λ = 620 nm (see Table 1).

_{inorg}. These relationships can be presented in the form of hyperbolic expressions (see Equations (39)–(42)). The statistical errors in the case of b

_{bp}as a function of two independent variables range from about 30% to about 50%, and they depend on the wavelength (see Table 1 and Table 2). Their magnitude is lower than magnitude of statistical errors in the case of b

_{bp}as a function of one independent variable.

_{rs}(λ

_{i}) in the waters of the southern Baltic can be described with satisfactory accuracy using the five-parameter model presented in this paper, the parameters in which are Chl a, ∑C, SPM, SPM

_{inorg}, and a

_{CDOM}(400). In this case, the statistical errors do not exceed 50% for all wavelengths, about 35% for λ = 420 nm and 488 nm, about 38% for λ = 555 nm and about 38% for λ = 420 nm (see Table 4).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Block diagram of the five-parameter model of R

_{rs}in the selected areas of the southern Baltic for four wavelengths (λ

_{i}= 420 nm, 488 nm, 555 nm, and 620 nm).

**Figure 3.**Relationships between (

**a**) a

_{ph}and Chl a, (

**b**) a

_{d}and SPM, (

**c**) a

_{CDOM}and a

_{CDOM}(400), (

**d**) b

_{bp}and SPM (for a wavelength of 488 nm) in the selected areas of the southern Baltic [line approximation by equations (

**a**) 8, (

**b**) 12, (

**c**) 20, and (

**d**) 16].

**Figure 4.**Ternary plots of the relative contribution of CDOM (a

_{CDOM}), non-phytoplankton pigments (a

_{d}) and phytoplankton pigments (a

_{ph}) to total absorption by non-water constituents at four wavelengths in the selected areas of the southern Baltic. The relative contribution of a given component was calculated as the ratio of the absorption coefficient of that component (e.g., a

_{ph}(λ

_{i})) and the sum of the absorption coefficients of all three components [a

_{CDOM}(λ

_{i}) + a

_{d}(λ

_{i}) + a

_{ph}(λ

_{i})].

**Figure 5.**Comparison of: a

_{ph}(λ

_{i}), a

_{d}(λ

_{i}), a

_{CDOM}(λ

_{i}) and b

_{bp}(λ

_{i}) measured (X(λ

_{i})

_{m}) and calculated (X(λ

_{i})

_{cal}) (

**a**–

**d**) using algorithms: (

**a**) Equation (8), (

**b**) Equation (12), (

**c**) Equation (20) and (

**d**) Equation (16) for a single wavelength (488 nm)-in the chosen areas of the southern Baltic. The solid line represents the function (X(λ

_{i})

_{m}= X(λ

_{i})

_{cal}). The probability density distribution of the ratio of calculated (X(λ

_{i})

_{cal}) to measured (X(λ

_{i})

_{m}) light absorption and backscattering coefficients (

**e**–

**h**) for wavelength 488 nm.

**Figure 6.**Comparison of: (

**a**) a

_{ph}calculated using Equation (8) and the ratio of Chl a and ∑C, (

**b**) a

_{d}calculated using Equation (12) and the ratio of SPM and SPM

_{inorg}, (

**c**) b

_{bp}calculated using Equation (16) and the ratio of SPM and SPM

_{inorg}, (for wavelength of 488 nm) in the selected areas of the southern Baltic [the line approximation by equations (

**a**) 27 (

**b**) 28 and (

**c**) 29] The coefficient of determination (R

^{2}): (

**a**) 0.27, (

**b**) 0.02, (

**c**) 0.04.

**Figure 7.**Comparison of: a

_{ph}(λ

_{i}), a

_{d}(λ

_{i}), and b

_{bp}(λ

_{i}) measured (X(λ

_{i})

_{m}) and calculated (X(λ

_{i})

_{cal}) (

**a**–

**c**) using algorithms: (

**a**) Equation (32), (

**b**) Equation (36), (

**c**) Equation (40) for a single wavelength (488 nm) in the chosen areas of the southern Baltic. The solid line represents the function (X(λ

_{i})

_{m}= X(λ

_{i})

_{cal}). The probability density distribution of the ratio of calculated light absorption and backscattering coefficients (X(λ

_{i})

_{cal}) to measured (X(λ

_{i})

_{m}) (

**d**–

**f**) for a single wavelength (488 nm). The coefficient of determination (R

^{2}): (

**a**) 0.92, (

**b**) 0.49, (

**c**) 0.87.

**Figure 8.**(

**a**–

**d**) Comparison of the measured R

_{rs}(λ

_{i})

_{m}and calculated R

_{rs}(λ

_{i})

_{cal}(Equations (43)–(46), Table 3) for four wavelengths (λ

_{i}= 420 nm, 488 nm, 555 nm and 620 nm) in the selected areas of the southern Baltic. The solid line represents the function (R

_{rs}(λ

_{i})

_{m}= R

_{rs}(λ

_{i})

_{cal}) (

**e**–

**h**). The probability density distribution of the ratio of calculated R

_{rs}(λ

_{i})

_{ca}to measured R

_{rs}(λ

_{i})

_{m}.

Step 1. 1-Parameter Model of IOPs | Arithmetic Statistic | Logarithmic Statistic | ||||
---|---|---|---|---|---|---|

Systematic Error | Statistical Error | Systematic Error | Standard Error Factor | Statistical Error | ||

<ε> [%] | σ_{ε} [%] | <ε>_{g} [%] | x | σ_{ε+} [%] | σ_{ε−} [%] | |

a_{ph}(420) Equation (7) | 4.39 | 30.97 | −1.54 × 10 ^{−2} | 1.35 | 34.66 | −25.74 |

a_{ph}(488) Equation (8) | 4.91 | 31.10 | 7.1 × 10 ^{−3} | 1.38 | 37.97 | −27.52 |

a_{ph}(555) Equation (9) | 4.46 | 30.80 | 3.3 × 10 ^{−3} | 1.35 | 35.01 | −25.93 |

a_{ph}(620) Equation (10) | 5.46 | 34.01 | 1.1 × 10 ^{−3} | 1.40 | 39.90 | −28.52 |

a_{d}(420) Equation (11) | 23.7 | 86.1 | −1.4 × 10 ^{−3} | 1.94 | 94.3 | −48.5 |

a_{d}(488) Equation (12) | 26.4 | 89.7 | −5.9 × 10 ^{−3} | 2.03 | 103.2 | −50.8 |

a_{d}(555) Equation (13) | 38.1 | 115.9 | 2.7 × 10 ^{−3} | 2.30 | 130.4 | −56.6 |

a_{d}(620) Equation (14) | 65.5 | 188.9 | 9.1 × 10 ^{−3} | 2.79 | 179.1 | −64.2 |

a_{CDOM}(420) Equation (19) | 0.23 | 7.08 | −0.01 | 1.07 | 7.25 | −6.76 |

a_{CDOM}(488) Equation (20) | 1.69 | 18.60 | 0.01 | 1.20 | 20.31 | −16.88 |

a_{CDOM}(555) Equation (21) | 4.71 | 32.91 | 0.01 | 1.35 | 35.37 | −26.13 |

a_{CDOM}(620) Equation (22) | 6.94 | 40.23 | 0.01 | 1.45 | 44.52 | −30.81 |

b_{bp}(420) Equation (15) | 8.87 | 44.52 | 8.25 × 10 ^{−3} | 1.53 | 52.93 | −34.61 |

b_{bp}(488) Equation (16) | 7.16 | 37.44 | 2.8 × 10 ^{−4} | 1.48 | 48.39 | −32.61 |

b_{bp}(555) Equation (17) | 6.97 | 36.14 | −2.22 × 10 ^{−3} | 1.48 | 48.20 | −32.52 |

b_{bp}(620) Equation (18) | 9.95 | 45.88 | 2 × 10 ^{−5} | 1.59 | 58.90 | −37.07 |

Step 2. 2-Parameter Model of IOPs | Arithmetic Statistic | Logarithmic Statistic | ||||
---|---|---|---|---|---|---|

Systematic Error | Statistical Error | Systematic Error | Standard Error Factor | Statistical Error | ||

<ε> [%] | σ_{ε} [%] | <ε>_{g} [%] | x | σ_{ε+} [%] | σ_{ε−} [%] | |

a_{ph}(420) Equation (31) | 3.86 | 28.90 | 6 × 10 ^{−4} | 1.32 | 32.16 | −24.33 |

a_{ph}(488) Equation (32) | 3.62 | 27.28 | −2 × 10 ^{−4} | 1.32 | 31.56 | −23.99 |

a_{ph}(555) Equation (33) | 4.21 | 29.84 | 1.3 × 10 ^{−3} | 1.34 | 33.96 | −25.35 |

a_{ph}(620) Equation (34) | 5.37 | 33.86 | −8 × 10 ^{−4} | 1.39 | 39.35 | −28.23 |

a_{d}(420) Equation (35) | 23.5 | 88.9 | −2.2 × 10 ^{−3} | 1.91 | 91.1 | −47.7 |

a_{d}(488) Equation (36) | 25.9 | 91.7 | 7 × 10 ^{−4} | 1.99 | 98.6 | −49.7 |

a_{d}(555) Equation (37) | 36.3 | 114.1 | −1.6 × 10 ^{−3} | 2.23 | 123.2 | −55.2 |

a_{d}(620) Equation (38) | 58.0 | 165.8 | 2 × 10 ^{−4} | 2.67 | 166.7 | −62.5 |

b_{bp}(420) Equation (39) | 8.86 | 45.31 | 7 × 10 ^{−4} | 1.52 | 52.26 | −34.32 |

b_{bp}(488) Equation (40) | 6.55 | 38.07 | 3 × 10 ^{−4} | 1.44 | 44.04 | −30.58 |

b_{bp}(555) Equation (41) | 6.02 | 36.19 | −4.4 × 10 ^{−3} | 1.42 | 42.04 | −29.60 |

b_{bp}(620) Equation (42) | 8.21 | 43.85 | −1.2 × 10 ^{−3} | 1.50 | 50.01 | −33.34 |

λ | C | B | D | K | J | L | a_{w} | b_{bw} |
---|---|---|---|---|---|---|---|---|

420 | 0.009 | 0.911 | 0.337 | 0.057 | 0.807 | 0.750 | 0.0045 | 0.0023 |

488 | 0.006 | 0.891 | 0.827 | 0.035 | 0.762 | 0.903 | 0.0147 | 0.0012 |

555 | 0.005 | 0.935 | 0.977 | 0.022 | 0.646 | 1.157 | 0.0596 | 0.0007 |

620 | 0.004 | 0.881 | 1.230 | 0.015 | 0.592 | 1.542 | 0.2755 | 0.0004 |

λ | F | G | H | M | N | P | f/Q | |

420 | 0.827 | 0.041 | 0.493 | 0.077 | 1.006 | 0.132 | 0.07 | |

488 | 0.820 | 0.022 | 0.824 | 0.624 | 1.077 | 0.485 | 0.10 | |

555 | 0.815 | 0.011 | 0.257 | 1.037 | 1.072 | 0.689 | 0.12 | |

620 | 0.926 | 0.007 | 0.261 | 1.488 | 1.136 | 0.794 | 0.13 |

Step 3. 5-Parameter Model of R_{rs}(λ_{i}) | Arithmetic Statistic | Logarithmic Statistic | ||||
---|---|---|---|---|---|---|

Systematic Error | Statistical Error | Systematic Error | Standard Error Factor | Statistical Error | ||

<ε> [%] | σ_{ε} [%] | <ε>_{g} [%] | x | σ_{ε+} [%] | σ_{ε−} [%] | |

R_{rs}(420) | 11.10 | 34.71 | 6.13 | 1.35 | 35.49 | −26.19 |

R_{rs}(488) | 10.16 | 34.53 | 5.46 | 1.34 | 34.13 | −25.44 |

R_{rs}(555) | 11.68 | 37.98 | 5.,94 | 1.38 | 38.25 | −27.67 |

R_{rs}(620) | 13.62 | 47.52 | 5.78 | 1.45 | 45.12 | −31.09 |

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**MDPI and ACS Style**

Lednicka, B.; Kubacka, M.
Semi-Empirical Model of Remote-Sensing Reflectance for Chosen Areas of the Southern Baltic. *Sensors* **2022**, *22*, 1105.
https://doi.org/10.3390/s22031105

**AMA Style**

Lednicka B, Kubacka M.
Semi-Empirical Model of Remote-Sensing Reflectance for Chosen Areas of the Southern Baltic. *Sensors*. 2022; 22(3):1105.
https://doi.org/10.3390/s22031105

**Chicago/Turabian Style**

Lednicka, Barbara, and Maria Kubacka.
2022. "Semi-Empirical Model of Remote-Sensing Reflectance for Chosen Areas of the Southern Baltic" *Sensors* 22, no. 3: 1105.
https://doi.org/10.3390/s22031105